21,603 research outputs found
Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing
Branching processes are a class of continuous-time Markov chains (CTMCs) with
ubiquitous applications. A general difficulty in statistical inference under
partially observed CTMC models arises in computing transition probabilities
when the discrete state space is large or uncountable. Classical methods such
as matrix exponentiation are infeasible for large or countably infinite state
spaces, and sampling-based alternatives are computationally intensive,
requiring a large integration step to impute over all possible hidden events.
Recent work has successfully applied generating function techniques to
computing transition probabilities for linear multitype branching processes.
While these techniques often require significantly fewer computations than
matrix exponentiation, they also become prohibitive in applications with large
populations. We propose a compressed sensing framework that significantly
accelerates the generating function method, decreasing computational cost up to
a logarithmic factor by only assuming the probability mass of transitions is
sparse. We demonstrate accurate and efficient transition probability
computations in branching process models for hematopoiesis and transposable
element evolution.Comment: 18 pages, 4 figures, 2 table
Matrix geometric approach for random walks: stability condition and equilibrium distribution
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest
neighbour (simple) random walk restricted on the lattice using the matrix
geometric approach. In particular, we first present an alternative approach for
the calculation of the stability condition, extending the result of Neuts drift
conditions [30] and connecting it with the result of Fayolle et al. which is
based on Lyapunov functions [13]. Furthermore, we consider the sub-class of
random walks with equilibrium distributions given as series of product-forms
and, for this class of random walks, we calculate the eigenvalues and the
corresponding eigenvectors of the infinite matrix appearing in the
matrix geometric approach. This result is obtained by connecting and extending
three existing approaches available for such an analysis: the matrix geometric
approach, the compensation approach and the boundary value problem method. In
this paper, we also present the spectral properties of the infinite matrix
A stochastic spectral analysis of transcriptional regulatory cascades
The past decade has seen great advances in our understanding of the role of
noise in gene regulation and the physical limits to signaling in biological
networks. Here we introduce the spectral method for computation of the joint
probability distribution over all species in a biological network. The spectral
method exploits the natural eigenfunctions of the master equation of
birth-death processes to solve for the joint distribution of modules within the
network, which then inform each other and facilitate calculation of the entire
joint distribution. We illustrate the method on a ubiquitous case in nature:
linear regulatory cascades. The efficiency of the method makes possible
numerical optimization of the input and regulatory parameters, revealing design
properties of, e.g., the most informative cascades. We find, for threshold
regulation, that a cascade of strong regulations converts a unimodal input to a
bimodal output, that multimodal inputs are no more informative than bimodal
inputs, and that a chain of up-regulations outperforms a chain of
down-regulations. We anticipate that this numerical approach may be useful for
modeling noise in a variety of small network topologies in biology
Multiphase modelling of vascular tumour growth in two spatial dimensions
In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters are investigated
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
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